989 research outputs found
Simulation based sequential Monte Carlo methods for discretely observed Markov processes
Parameter estimation for discretely observed Markov processes is a
challenging problem. However, simulation of Markov processes is straightforward
using the Gillespie algorithm. We exploit this ease of simulation to develop an
effective sequential Monte Carlo (SMC) algorithm for obtaining samples from the
posterior distribution of the parameters. In particular, we introduce two key
innovations, coupled simulations, which allow us to study multiple parameter
values on the basis of a single simulation, and a simple, yet effective,
importance sampling scheme for steering simulations towards the observed data.
These innovations substantially improve the efficiency of the SMC algorithm
with minimal effect on the speed of the simulation process. The SMC algorithm
is successfully applied to two examples, a Lotka-Volterra model and a
Repressilator model.Comment: 27 pages, 5 figure
Fenichel\u27s theorems with applications in dynamical systems.
Three main theorems due to Fenichel are fundamental tools in the exploration of geometric singular perturbation theory. This expository paper attempts to provide an introduction to the concepts stated in Fenichel\u27s theorems and provide illustrative examples. The goal is to provide enough insight to gain a basic understanding of the usefullness of these theorems
Finite-Time Stability Analysis and Control for a Class of Stochastic Singular Biological Economic Systems Based on T-S Fuzzy Model
This paper studies the problem of finite-time stability and control for a class of stochastic singular biological economic systems. It shows that such systems exhibit the distinct dynamic behavior when the economic profit is a variable rather than a constant. Firstly, the stochastic singular biological economic systems are established as fuzzy models based on T-S fuzzy control approach. These models are described by stochastic singular T-S fuzzy systems. Then, novel sufficient conditions of finite-time stability are obtained for the stochastic singular biological economic systems, and the state feedback controller is designed so that the population (state of the systems) can be driven to the bounded range by the management of the open resource. Finally, by using Matlab software, numerical examples are given to illustrate the effectiveness of the obtained results
Logistics of Mathematical Modeling-Focused Projects
This article addresses the logistics of implementing projects in an
undergraduate mathematics class and is intended both for new instructors and
for instructors who have had negative experiences implementing projects in the
past. Project implementation is given for both lower and upper division
mathematics courses with an emphasis on mathematical modeling and data
collection. Projects provide tangible connections to course content which can
motivate students to learn at a deeper level. Logistical pitfalls and insights
are highlighted as well as descriptions of several key implementation
resources. Effective assessment tools, which allowed me to smoothly adjust to
student feedback, are demonstrated for a sample class. As I smoothed the
transition into each project and guided students through the use of the
technology, their negative feedback on projects decreased and more students
noted how the projects had enhanced their understanding of the course topics.
Best practices learned over the years are given along with project summaries
and sample topics. These projects were implemented at a small liberal arts
university, but advice is given to extend them to larger classes for broader
use.Comment: 27 pages, no figures, 1 tabl
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